Abstract
The thermal field theory is applied to fermionic superfluids by doubling the degrees of freedom of the BCS theory. We construct the two-mode states and the corresponding Bogoliubov transformation to obtain the BCS thermal vacuum. The expectation values with respect to the BCS thermal vacuum produce the statistical average of the thermodynamic quantities. The BCS thermal vacuum allows a quantum-mechanical perturbation theory with the BCS theory serving as the unperturbed state. We evaluate the leading-order corrections to the order parameter and other physical quantities from the perturbation theory. A direct evaluation of the pairing correlation as a function of temperature shows the pseudogap phenomenon, where the pairing persists when the order parameter vanishes, emerges from the perturbation theory. The correspondence between the thermal vacuum and purification of the density matrix allows a unitary transformation, and we found the geometric phase associated with the transformation in the parameter space.
Highlights
Quantum many-body systems can be described by quantum field theories[1,2,3,4]
We will apply the thermal field theory to develop a perturbation theory where the BCS thermal vacuum serves as the unperturbed state, and the fermion-fermion interaction ignored in the BCS approximation is the perturbation
Since the thermal vacuum is a pure state, the statistical average of a physical quantity at finite temperatures is the expectation value obtained in the quantum mechanical manner
Summary
The thermal field theory is applied to fermionic superfluids by doubling the degrees of freedom of the BCS theory. Since the thermal vacuum is a pure state, the statistical average of a physical quantity at finite temperatures is the expectation value obtained in the quantum mechanical manner. The central idea of thermal field theory is to express the statistical average over a set of mixed quantum states as the expectation value of a temperature-dependent pure state, called the thermal vacuum |0(β)〉9,22. In zero-energy eigenstate noninteracting bosons of or fermions and the BCS superfluid, the thermal vacuum can be constructed by performing a unitary transformation U(β) on the two-mode ground state,|0(β)〉 = U(β)|0, 0 〉 via a Bogoliubov transformation. In those cases, the thermal vacuum is an eigenstate of the “thermal Hamiltonian” H(T) ≡ U(β)HU−1(β) with eigenvalue E0, which is the ground-state energy of H. Based on the BCS thermal vacuum, we will develop a perturbation theory for systems where U(β) cannot be constructed explicitly
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